To determine To solve: The ODE y ″ + 0.54 y ′ + ( 0.0729 + π ) y = 0 for the given initial values and check whether the answer satisfies the ODE also the initial conditions. Explanation Case (i): If the auxiliary equation has distinct real roots λ 1 , and λ 2 of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e λ 1 x , e λ 2 x and the general solution of y ″ + a y ′ + b y = 0 is y = c 1 e λ 1 x + c 2 e λ 2 x . Case (ii): If the auxiliary equation has real double root λ = − a of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e − a x , x e − a x and the general solution of y ″ + a y ′ + b y = 0 is y = ( c 1 + c 2 ) e − a x . Case (iii): If the auxiliary equation has complex conjugate root λ 1 = a + i ω , λ 2 = a − i ω of y = e λ x , then the basis of y ″ + a y ′ + b y = 0 is e a x cos ω x , e a x sin ω x and the general solution of y ″ + a y ′ + b y = 0 is y = e − a x ( A cos ω x + B sin ω x ) . Calculation: The given ODE is y ″ + 0.54 y ′ + ( 0.0729 + π ) y = 0 (1) Compute the auxiliary equation of the above ODE as follows. Let y = e λ x . That implies, the first and second derivative of y is y ′ = λ e λ x and y ″ = λ 2 e λ x , respectively. Substitute y ″ = λ 2 e λ x and y = e λ x in (1) and simplify further. λ 2 e λ x + 0.54 λ e λ x + ( 0.0729 + π ) e λ x = 0 e λ x ( λ 2 + 0.54 λ + ( 0.0729 + π ) ) = 0 λ 2 + 0.54 λ + ( 0.0729 + π ) = 0 That is, the characteristic equation is λ 2 + 0.54 λ + ( 0.0729 + π ) = 0 . Compute the roots of the above characteristic equation as follows. Substitute a = 1, b = 0.54 and c = 0.0729 + π in λ = − b ± b 2 − 4 a c 2 a . λ = − 0.54 ± ( 0.54 ) 2 − 4 ( 1 ) ( 0.0729 + π ) 2 ( 1 ) = − 0.54 ± 0.2916 − 0.2916 − 4 π 2 = − 0.54 ± 2 i π 2 = − 0.27 + i π or − 0.37 − i π From the above calculation, it is observed that y ″ + 0.54 y ′ + ( 0.0729 + π ) y = 0 has complex conjugate roots. Then by case (iii), the basis of y ″ + 0.54 y ′ + ( 0.0729 + π ) y = 0 is e − 0.27 x cos ( π x ) , e − 0.27 x sin ( π x ) . The general solution of y ″ + 0.54 y ′ + ( 0.0729 + π ) y = 0 is y = e − 0.27 x ( c 1 cos ( π x ) + c 2 sin ( π x ) ) . Now check the correctness of the above solution as follows. Compute the second derivative of y as follows. y ′ = { e − 0.27 x ( − 0.27 ) ( c 1 cos ( π x ) + c 2 sin ( π x ) ) + e − 0.27 x ( − c 1 sin ( π x ) ( π ) + c 2 cos ( π x ) ( π ) ) } = { − 0.27 e − 0.27 x ( c 1 cos ( π x ) + c 2 sin ( π x ) ) + e − 0.27 x ( − c 1 sin ( π x ) ( π ) + c 2 cos ( π x ) ( π ) ) } y ″ = { − 0.27 e − 0.27 x ( − 0.27 ) ( c 1 cos ( π x ) + c 2 sin ( π x ) ) − 0.27 e − 0.27 x ( − c 1 sin ( π x ) ( π ) + c 2 cos ( π x ) ( π ) ) + e − 0.27 x ( − 0.27 ) ( − π c 1 sin ( π x ) + π c 2 cos ( π x ) ) + e − 0.27 x ( − π c 1 cos ( π x ) − π c 2 sin ( π x ) ) } = { 0.0729 c 1 e − 0.27 x cos ( π x ) + 0.0729 c 2 e − 0.27 x sin ( π x ) + 0

Custom Kreyszig: Advanced Engineering Mathematics

10th Edition

ISBN: 9781119166856

Author: Kreyszig

Publisher: JOHN WILEY+SONS INC.CUSTOM

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 2.2, Problem 29P

To determine

To solve: The ODE y″+0.54y′+(0.0729+π)y=0 for the given initial values and check whether the answer satisfies the ODE also the initial conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.2, Problem 28P

Chapter 2.2, Problem 30P

Students have asked these similar questions

1 S 0 sin(lnx) x² - 1 Inx dx

2 6. Modelling. Suppose that we have two tanks (A and B) between which a mixture of brine flows. Tank A contains 200 liters of water in which 50 kilograms of salt has been dissolved and Tank B contains 100 liters of pure water. Water containing 1kg of salt per liter is pumped into Tank A at the rate of 5 liters per minute. Brine mixture is pumped into Tank A from Tank B at the rate of 3 liters per minute and brine mixture is pumped from Tank A into Tank B at the rate of 8 liters per minute. Brine is drained from Tank B at a rate of 5 liters per minute. (a) Draw and carefully label a picture of the situation, including both tanks and the flow of brine between them. JankA 1ks of Salt Slits Pump EL Brine mit tark A from tank 13 Tank 13 k 3L zooliters of Ico liters of water with pure water. Saky salt → 777 disslore inside Brine mix is pumped from tank A to B of 82 Brine drainen min by Gf salt (b) Assume all brine mixtures are well-stirred. If we let t be the time in minutes, let x(t) 1ks…

No chatgpt pls

Chapter 2 Solutions

Custom Kreyszig: Advanced Engineering Mathematics

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.